Math Problems : Practices
9:27 PM
Muhammad Yusuf
, Posted in
1. Two players, A and B, are playing the following game. They take turns writing down the digits of a six-digit number from left to right; A writes down the first digit, which must be nonzero, and repetition of digits is not permitted. Player A wins the game if the resulting six-digit number is divisible by 2, 3 or 5, and B wins otherwise.
Prove that A has a winning strategy.
2. Prove that nn - n is divisible by 24 for all odd positive integers n.
3. Let a and b be real numbers. Prove that the inequality
holds.
When does equality hold?
4. Let ABCD be a quadrilateral. The circumcircle of the triangle ABC intersects the sides CD and DA in the points P and Q respectively, while the circumcircle of CDA intersects the sides AB and BC in the points R and S. The straight lines BP and BQ intersect the straight line RS in the points M and N respectively. Prove that the points M, N, P and Q lie on the same circle.
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